Understanding how changes in the equation of a function result in
stretching
and/or
reflecting
the graph of the function is a great way to take some of the mystery out of graphing more complicated equations. By recognizing the family to which a more complex equation belongs, and then identifying what changes have been made to the parent of that family, the graph of even quite detailed functions can be made much more understandable.

See if you can identify what parts of the equation:
represent either a stretch or a reflection of the parent function:
before the review at the end of this lesson.

The graph of
y
= -
x
2
represents a
reflection
of
y
=
x
2
, over the
x
-axis. That is, every function value of
y
= -
x
2
is the negative of a function value of
y
=
x
2
. In general,
g
(
x
) = -
f
(
x
) has a graph that is the graph of
f
(
x
), reflected over the
x
-axis.

Example A

Identify the graph of the function
y
= (3
x
)
2
.

Solution

We have multiplied
x
by 3. This should affect the graph horizontally. However, if we simplify the equation, we get
y
= 9
x
2
. Therefore the graph if this parabola will be taller/thinner than
y
=
x
2
. Multiplying
x
by a number greater than 1 creates a horizontal compression, which looks like a vertical stretch.

Example B

Identify the transformation described by
y
= ((1/2)
x
)
2
.

Solution

If we simplify this equation, we get
y
= (1/4)
x
2
. Therefore multiplying
x
by a number between 0 and 1 creates a horizontal stretch, which looks like a vertical compression. That is, the parabola will be shorter/wider.

Example C

Sketch a graph of
y
=
x
3
and y = -
x
3
on the same axes.

Solution:

At first the two functions might look like two parabolas. If you graph by hand, or if you set your calculator to sequential mode (and not simultaneous), you can see that the graph of
y
= -
x
3
is in fact a reflection of
y
=
x
3
over the x-axis.

However, if you look at the graph, you can see that it is a reflection over the
y
-axis as well. This is the case because in order to obtain a reflection over the
y
-axis, we negate
x
. In other words,
h
(
x
) =
f
(-
x
) is a reflection of
f
(
x
) over the
y
-axis. For the function
y
=
x
3
,
h
(
x
) = (-
x
)
3
= (-
x
) (-
x
) (-
x
) = -
x
3
. This is the same function as the one we have already graphed.

It is important to note that this is a special case. The graph of
y
=
x
2
is also a special case. If we want to reflect
y
=
x
2
over the
y
-axis, we will just get the same graph! This can be explained algebraically:
y
= (-
x
)
2
= (-
x
) (-
x
) =
x
2
.

Concept question wrap-up:

Are you able to identify the transformations described in the beginning of the lesson now?

The function:
is the result of transforming
by:

reflecting it over the
x
axis, because of the negative co-efficient on the
x
. and:

vertically compressing it (making it
wider
), because of the co-efficient being a fraction between 0 and 1.

Vocabulary

Reflections
are transformations which result in a "mirror image" of a parent function. They are a result of differing signs between parent and child functions.

Stretches
are transformations which result in the width of a graph being increased or decreased. They are the result of the co-efficient of the
x
term being between 0 and 1.

Guided Practice

Questions

1) Graph the functions
and
.

2) Sketch the graph of
by appropriately stretching the parent graph

3) Sketch the graph of
by reflecting the graph of
above.

4) Sketch the graph of
by appropriately stretching

5) Identify the function and sketch the graph of
reflected over both axes

Answers

1) The equation
might look confusing because of the -
x
under the square root. It is important to keep in mind that -
x
means the
opposite
of
x
. Therefore the domain of this function is restricted to values ≤ 0. For example, if
x
= - 4,
. It is this domain, which includes all real numbers not in the domain of
plus zero, that gives us a graph that is a reflection over the
y
-axis.

In sum, a graph represents a reflection over the
x
-axis if the function has been negated (i.e. the
y
has been negated if we think of
y
=
f
(
x
)). The graph represents a reflection over the
y
-axis if the variable
x
has been negated.

2) The graph of
is the graph of the parent,
, with each y-coordinate multiplied by 3. The image below shows both the parent and the child function on the same axes.

3) The graph of
is the graph of
reflected over the x-axis, the image below shows both functions.

4) The graph of
is the graph of
with each co-ordinate multiplied by 3, the image below shows both graphs.

5) To reflect the graph of
over both axes, the function must be
negated
both outside and inside the root:
. The negation (negative)
outside
of the root has the effect of reflecting the graph vertically, and the negation
inside
of the root reflects the graph horizontally. The image below shows three versions:

a) (BLUE)

b) (GREEN)

c) (RED)

Practice

If a function is multiplied by a coefficient, what will happen to the graph of the function?

What does multiplying x by a number greater than one create?

What happens when we multiply x by a number between 0 and 1

In order to obtain a reflection over the y axis what do we have to do to x?

how do we obtain a reflection over the x-axis?

Write a function that will create a horizontal compression of the following:

Write a function that will horizontally stretch the following:

Rewrite this function
to get a reflection over the x-axis.

Rewrite this function
to get a reflection over the y-axis.

Graph each of the following using transformations. Identify the translations and reflections.

Let
be the function defined by the line segment connecting the points (-1, 4) and (2, 5). Graph each of the following transformations of
.

The graph of y = x is shown below. Sketch the graph of each of the following transformations of y = x

Image Attributions

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Description

Learning Objectives

Here you will explore how to take the
parent function
in a
function family
and either
stretch
it horizontally or vertically, or
reflect
it to change its direction. If you are not already familiar with
function families
, it would be a good idea to review them first, and then return here to apply those concepts.